The standard way of revealing the breakdown of the replica-symmetric solution is to perform a stability analysis
following the pattern of the famous de Almeida-Thouless paper [1]} in the theory of spin glasses, i.e.
magnetic systems with random interactions.
Such analysis can be straightforwardly done for the present type of system, see Appendix D of the present lectures,
and shows that for a given value of \(R\) 
the replica symmetric solution becomes unstable for the temperatures \(T<T_c=g\frac{R^2-1}{R^2+1}\) .
Therefore at low temperatures stable solution will have to be one with a broken symmetry in the replica space.
To derive the corresponding expression for the free energy of our model we will follow a particular heuristic scheme of the
replica symmetry breaking proposed originally by Parisi in the theory of spin glasses, see e.g. [2]}, or more recently
[3]} In recent years the use of the scheme was justified by alternative rigorous mathematical procedures.
For the model under consideration the corresponding equations
were re-derived recently by a rigorous methods in [4]} without any recourse to
the powerful but ill-defined replica trick.. To make the present set of lectures self-contained
we describe in full detail the structure of the matrix \(Q\) , the ensuing Parisi function \(x(q)\)  and the main steps of the derivation
in Appendix C in full detailThe Appendix is taken verbatim from [5]}, but we
used this opportunity to correct the important formula (REF ) which appeared in [5]} in a distorted form..
Here we just sketch those objects schematically for the convenience of the reader:
